Facebook only notifies you of messages from your friends, and yet anyone can send you a message on…
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Blogger Erika Napoletano, who discovered the "other" inbox a few days ago, notes that you can move any message to your "main" inbox with the click of a button. From there, you'll get notifications from that user where you didn't previously. You can't make Facebook do this automatically for all messages—so you'll still have to check that "Other" inbox regularly—but this will ensure that when you start an important conversation, you get notifications for the rest of the messages that come from that person later on.

To move a message, just go to your "Other" inbox and find a conversation for which you want to start receiving notifications. Then, click the "Actions" button and press "Move to Messages". From then on, that thread will be in your normal Messages inbox and you should get notifications normally.

It's hardly a perfect fix, but until Facebook gets their act in gear, it can at least keep you abreast of the important messages you find in your second inbox. If you know of a better fix though, by all means, let us know about it in the comments below.